Linear Spaces

Linear (Vector) Spaces

Definition: A linear space (or vector space) over a field FF is a set VV with two operations, addition and scalar multiplication such that the following axioms hold:

Vector Addition (+):V×V→V(+): V \times V \rightarrow V such that x+y∈Vx + y \in V for all x,y∈Vx,y \in V.

Scalar Multiplication (⋅):F×V→V(\cdot): F \times V \rightarrow V such that αx∈V\alpha x \in V for all α∈F\alpha \in F and x∈Vx \in V.

Elements of the vector space are called vectors, or points.

Scalar multiplication in a vector space depends on the field FF. Thus when we say x∈Vx \in V, we mean that xx is a vector in the vector space VV over the field FF or (V,F)(V,F).
For example, Rn\mathbb{R}^n is a vector space over the field R\mathbb{R}, and Cn\mathbb{C}^n is a vector space over the field C\mathbb{C}.

Sharing the same field is a necessary condition for two vector spaces to be comparable. For example, Rn\mathbb{R}^n and Cn\mathbb{C}^n are not comparable.

The simplest vector space contains only one point. In other words, {0}\{0\} is a vector space.


Example: Show that x0F=0Vx 0_F = 0_V for all x∈Vx \in V
Proof:

  1. y=x0F=0Vy = x0_F = 0_V is assumed to be true.
  2. x+y=x+x0F=x1F+x0Fx + y = x + x0_F = x1_F+ x0_F (M4)
  3. x+y=x(1F+0F)=x1Fx + y = x(1_F + 0_F) = x1_F (M3)
  4. x+y+(−x)=x1F+(−x1F)x + y + (-x) = x1_F + (-x1_F) (A4)
  5. y=x(1F+(−1F))=x0V=0Vy = x(1_F + (-1_F)) = x0_V = 0_V (A2) ■\blacksquare

Example: Show that x0F=0vx0_F = 0_v
Proof:

  1. Assume y=x0Fy = x0_F and show y=0Vy = 0_V.
  2. x+y=x+x0F=x1F+x0Fx + y = x + x0_F = x1_F + x0_F (M4)
  3. x+y=x1F+x0F=x(1F+0F)x + y = x1_F + x0_F = x(1_F + 0_F) (M3)
  4. x+y=x(1F+0F)=x1Fx + y = x(1_F + 0_F) = x1_F (A3)
  5. x+y=x1F=xx + y = x1_F = x (M4)
  6. x+y+(−x)=x+(−x)x + y + (-x) = x + (-x) (A4)
  7. y=x+(−x)=0Vy = x + (-x) = 0_V (A4) ■\blacksquare

Remark: A vector space has a unique additive identity.

Function Space

Definition: Let SS be a set and FF be a field. The set of all functions from SS to FF is denoted by FSF^S.

For f,g∈FSf,g \in F^S, the sum f+g∈FSf+g \in F^S is the function defined by

(f+g)(x)=f(x)+g(x), ∀x∈S(f+g)(x) = f(x) + g(x), \ \forall x \in S

For f∈FSf \in F^S and α∈F\alpha \in F, the product αf∈FS\alpha f \in F^S is the function defined by

(αf)(x)=αf(x), ∀x∈S(\alpha f)(x) = \alpha f(x), \ \forall x \in S

Example: Set of all polynomials with degree n with coefficients in FF is denoted by F[x]nF[x]_n.

What essentially function spaces are, is that they are the set of all functions from a set to a field. For example, the set of all polynomials with degree n with coefficients in FF is denoted by F[x]nF[x]_n.


#EE501 - Linear Systems Theory at METU